Re: a chair problem

doraeyee_u_v,I think the question can be reduced to showing four points in a three dimensional space are coplanar.I think this can be done mathematically by forming the equation of the plane with the help of the coordinates of three points and showing that the fourth point lies in the plane.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Re: a chair problem

If the seat is flat and the legs are the same length then there exists room to make a perfect chair and an imperfect chair.There are many scenarios with the legs at various angles that would still work, but perfectly vertical legs would be something to prove. Make vertical the y-axis and flat across the front of the chair the x-axis. We know the bottom of the seat where the legs come out is perfectly flat, so we say the the y-values are all the same, hence the slope or rise over run on the seat is zero.Zero = 0 vertical change / some horizontal distance.Now the legs are vertical, so the x values do not change for the calculations of the bottom of the legs, only the y values.The y values all are reduced by the same value, the leg length, so all of the y values stay the same for the bottom of the legs.Therefore if you do slope calculations between any of the six pairs of legs choosable, they will all have a slope of zero again.